- Today we review. Maybe you've brought some questions....
- Regarding IMath assignments: you can still take me up on my offer
to re-open IMath assignments.
Email me with your requests for any that you want to try (tell me which assignments you want re-opened).
NB: Please don't pay any attention to your percentage score on IMath. Each assignment is equally weighted.
- Our exam is Wednesday, 2:30-4:30.
- It will be a two-hour, cumulative exam.
- It will be about 60% new (since the midterm) and 40% old material.
- You will be allowed a page of notes, front and back.
- You are allowed a calculator.
- You might check the daily agendas for the reviews of our previous exam material. Certainly look over the midterm exam, quizzes, and keys.
- You will be allowed to skip some questions, but not all questions; there will be some questions which you may not skip.
- Several questions will look like problems right off the quizzes or IMath.
- Our new (and final) topic was Section
3.4: Applied Optimization.
- We showed that dogs do calculus (even if they can't differentiate
even the simplest polynomials).
- Our authors suggest a strategy for solving these problems.
- We started a 3.4 worksheet? Your last one...
Let's begin by finishing that problem I was working at the board as class ended.
Then we'll see if you have any other questions about either of the two optimization worksheets (including the 3.3 worksheet). Here are keys for both:
- We showed that dogs do calculus (even if they can't differentiate
even the simplest polynomials).
- We review; and let's begin with questions that you may have.
- We began by discussing "linear things" -- things well represented
by a ruler.
We learned that the telltale sign of a linear relationship is that if you double one quantity (call it \(x\)), then you'll double the other (call it \(y\)):
\[ y=mx \]
- The crucial thing is the slope, \(m\): it tells us how \(y\) is
changing as \(x\) changes: increasing or decreasing or constant
(\(y=0\)), and by how much, how rapidly.
- When things aren't truly linear (but only "microscopically
linear", as you zoom in on a curve), then we need a new concept
-- we call that concept the derivative.
- Let me remind you of the most important definition in calculus,
and that it is built on an indeterminate form.
\[ f'(a)=\lim_{h\to 0}{\frac{f(a+h)-f(a)}{h}} \]
This animal measures the instantaneous rate of change of the function $f$; and it turns out that it's really useful for finding extrema of functions (places where the rate of change is zero!), in particular.
- Rates of change are everywhere: Trees
are moving north from global warming. Look up how your city could
change. (Washington Post). (Derivatives can be extended to higher dimensions!)
- Along the way we have spent some time talking about the derivative
in the context of data: sometimes you don't have a
formula. In this case, you have to either work with
approximations (average rates), or model the data with a
function (which we saw, for example, in the context of cherry
blossoming times).
- And while the limit definition is the thing on which all of
calculus is based, the good news is that there are lots of
rules that make our lives easier when it comes to calculating
derivatives, of all of our favorite functions.
Although you may have a sheet with all of the rules on it (which I suggest!), you want to be able to use them fluidly, and demonstrate how they work.
You might even be called upon to show how we derive the rule from the limit definition (e.g. the sum rule, or the product rule).
- I've prepared a "sample final"; it will give you some idea of the kinds of problems you might encounter.
- Do Dogs Know Calculus?
- Cherry
blossoming data (from the EPA). Some "Key Points" cited in that webpage:
- Based on the entire 102 years of data, the average peak bloom date for Washington's cherry blossoms is April 4.
- Peak bloom date for the cherry trees is occurring earlier than it did in the past. Since 1921, peak bloom dates have shifted earlier by approximately seven days. The peak bloom date has occurred before April 4 in 16 of the past 20 years.
- While the length of the National Cherry Blossom Festival has continued to expand, the Yoshino cherry trees have bloomed near the beginning of the festival in recent years. During some years, the festival missed early peak bloom dates entirely.
- More generally, Leaf and Bloom Dates (EPA)